Semiparametric analysis of isotonic errors-in-variables regression models with randomly right censored response

Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, the Weighted Least Square method and a pivotal quantity method were used for unknown population variance cases. The result of the theoretical study was that pivotal quantity distributed student-t. The shortest parameter interval estimation of semi parametric spline truncated regression model was obtained through the optimization process using the method of Lagrange.Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, t...

ملخص:- تحتوي نماذج الانحدار على مکونين أحدهما مکون عشوائي يمثل حد الخطأ (Error Term) والأخر مکون غير عشوائي يمثل متغير الاستجابة(Response Variable) وهذا المکون هو محل الاهتمام فيتم تقديره بإحدى الطريقتين هما: الطريقة المعلمية أو الطريقة اللامعلمية أو کلاهما. نموذج الانحدار المعلمي (Parametric Regression Model) هو النموذج الأکثر شيوعا حيث أنه يفترض أن متغير الاستجابة له صيغه دالية محدده مسبقا والتي تم الحصول عليها من معلومات سابقه حول هيکل دالة الانحدار من خلال ذلک يتم تقدير فئة من المعالم. ورغم أن النماذج المعلمية تعد أحد أهم أدوات تحليل البيانات؛ لأنها فعاله ويمکن تفسيرها بسهوله، إلا انها لم تستطع ان تکون کافية في بعض الحالات. أما نموذج الانحدار اللامعلمي (Nonparametric Regression Model) فانه يعتمد على تقدير دالة الانحدار مباشرة من البيانات کما أنه لا يفترض فروضا معينه لتقدير معالم النموذج مثل التي يفترضها النموذج المعلمي وبالتالي فهو يتمتع بمرونة أکثر لاکتشاف البيانات التي قد تکون مفقودة او في الحالات التي لا تتوافر فيها معلومات سابقة. يتم انشاء نموذج الانحدار شبه المعلمي (Semi parametric Regression Model) من خلال مزج الطريقتين السابقتين معا. فغالبا ما يتم استخدام هذا النموذج في المواقف التي تکون فيها الافتراضات المعلمية غير محددة وغير متسقة، أو أن النموذج اللامعلمي لا يعمل بشکل کامل. يشتمل هذا البحث على تطبيق نماذج الانحدار المعلمية وشبه المعلمية بالإضافة إلى النموذج اللامعلمى کحالة خاصة من نموذج الانحدار شبه المعلمى لتقدير الدالة الافتراضية وتحديد شکل العلاقة بين المتغير التابع والمتغيرات المستقلة بالاضافة الى شکل انتشار البيانات مع خط (أو منحني) التقدير وذلک على بيانات واقعية تم الحصول عليها من الإتحاد المصري لألعاب القوى والمعروفة بـبيانات لعبة العشاري . Regression models contain two components, one of which is a random component that represents the Error Term and the other is a non-random component that represents the response variable and this component is of interest and is estimated in one of two methods: the parametric method or the nonparametric method or both. Parametric regression model is the most common model since it assumes that the response variable has a predetermined functional formula which was obtained from previous information about the structure of the regression function by which a class of parameters is estimated. Although parameter models are one of the most important tools for analyzing data; Because it is effective and can be explained easily, but it could not be sufficient in some cases. As for the nonparametric regression model, it relies on estimating the regression function directly from the data, as it does not assume specific assumptions for estimating the model parameters, such as those assumed by the parametric model, and thus it has more flexibility to discover data that may be missing or in cases where previous information is not available. The semi-parametric regression model is constructed by blending the two previous methods together. This model is often used in situations where the parametric assumptions are indeterminate and inconsistent, or the nonparametric model is not fully functional. This research includes the application of parametric and semi-parametric regression models in addition to the nonparametric model as a special case of the semi-parametric regression model for estimating the hypothetical function and determining the form of the relationship between the dependent variable and the independent variables in addition to the data spread form with a line (or curve) estimation on realistic data On it from the Egyptian Athletics Federation, known as Decathlon data.

Binary logistic semiparametric regression analysis is a commonly used statistical technique when the dependent variable is dichotomous or binary. In this analysis, the relationship between the success probability and certain explanatory variables is assumed to have a linear form, while the relationship to other variables is unknown. Multicollinearity is a serious problem that arises when explanatory variables in logistic semiparametric regression are highly correlated. It is well known that the variance of the maximum likelihood estimator is inflated due to multicollinearity in the semiparametric logistic regression model. Therefore, a novel stochastic restricted iterative weighted ridge estimator for logistic semiparametric regression is introduced, and its statistical properties are extracted asymptotically. Moreover, an extension of the generalized cross validation (GCV) function is introduced and applied for choosing the best values of the ridge parameter and the bandwidth of the kernel smoother. Additionally, some theorems are developed to illustrate the convergence of the GCV mean. Ultimately, the Monte-Carlo simulation studies and an actual real-life data set are conducted to support our theoretical discussion, and the findings indicated that the new estimator outperformed the other estimators under consideration.

Root-n-consistent and efficient estimation in semiparametric additive regression models

This dissertation consists of two chapters: Chapter 1 develops nonparametric and semiparametric regression methodologies which relate the group testing responses to the individual covariates information. In this chapter, we extend the parametric regression model of Xie (2001) for binary group testing data to the nonparametric and semiparametric models. We fit nonparametric and semiparametric models and obtain estimators of the parameters by maximizing penalized likelihood function. For implementation, we apply EM algorithm considering the individual responses as complete data and the group testing responses as observed data. Simulation studies are performed to illustrate the methodologies and to evaluate the finite sample performance of our methods. In general, group testing involves a large number of subjects, hence, the computational aspect is also discussed. The results show that our estimation methods perform well for estimating both the individual probability of positive outcome and the prevalence rate in the population.Chapter 2 studies a partially linear regression model with missing response variable and develops semiparametric efficient inference for the parametric component of the model. The missingness considered here includes a broad range of missing patterns. For the estimation method, we use the concept of least favorable curve, least favorable direction and the generalized profile likelihood in Severini and Wong (1992). Asymptotic distributions for the estimators of the parametric components are obtained. It is shown that the estimators are asymptotically normally distributed under some conditions. Furthermore, we prove that the asymptotic covariance of the estimators achieves the semiparametric lower bound under the regularity conditions and additional conditions given in the appendix.We also propose an algorithm which runs iteratively between fitting parametric components and fitting nonparametric components while holding the other fixed. EM algorithms are used in estimating the parametric components by a semiparametric estimating equation and in estimating the nonparametric components by smoothing methods. It is proved that the estimators from this iterative algorithm equal to the conditional expectations (conditioned on observed data) of the semiparametric efficient estimators from complete data. The methodology is illustrated and evaluated by numerical examples.

Sequential estimation for time series regression models

Size-biased sampling arises when a positive-valued outcome variable is sampled with selection probability proportional to its size. In this article, we propose a semiparametric linear regression model to analyze size-biased outcomes. In our proposed model, the regression parameters of covariates are of major interest, while the distribution of random errors is unspecified. Under the proposed model, we discover that regression parameters are invariant regardless of size-biased sampling. Following this invariance property, we develop a simple estimation procedure for inferences. Our proposed methods are evaluated in simulation studies and applied to two real data analyses.

In this paper, estimation of the linear regression\nmodel is made by ordinary least squares method and the\npartially linear regression model is estimated by penalized\nleast squares method using smoothing spline. Then, it is\ninvestigated that differences and similarity in the sum of\nsquares related for linear regression and partial linear\nregression models (semi-parametric regression models). It is\ndenoted that the sum of squares in linear regression is reduced\nto sum of squares in partial linear regression models.\nFurthermore, we indicated that various sums of squares in the\nlinear regression are similar to different deviance statements in\npartial linear regression. In addition to, coefficient of the\ndetermination derived in linear regression model is easily\ngeneralized to coefficient of the determination of the partial\nlinear regression model. For this aim, it is made two different\napplications. A simulated and a real data set are considered to\nprove the claim mentioned here. In this way, this study is\nsupported with a simulation and a real data example.

The problem of fitting a proportional odds regression model to data from the case-cohort design proposed by Prentice is considered. A weighted semiparametric likelihood method is proposed. Under the proportional odds model, the maximum weighted-semiparametric likelihood estimators of both the regression parameter and the transformation function are shown to be consistent and normally distributed. The applicability of the weighted semiparametric likelihood method to the semiparametric transformation regression models is also discussed. In particular, when the proportional hazards regression model is fitted, estimators proposed by Chen and Lo can be generated by the weighted semiparametric likelihood method under different weighting schemes. A simulation study suggests that the case-cohort design is also useful under the proportional odds regression model and the proposed method performs well with practical finite sample sizes.

This paper deals with the construction of efficient estimators in semiparametric models without the sample splitting technique. Schick (1987) gave sufficient conditions using the leave-one-out technique for a construction without sample splitting. His conditions are stronger and more cumbersome to verify than the necessary and sufficient conditions for the existence of efficient estimators which suffice for the construction based on sample splitting. In this paper we use a conditioning argument to weaken Schick′s conditions. We shall then show that in a large class of semiparametric models and for properly chosen estimators of the score function the resulting weaker conditions reduce to the minimal conditions for the construction with sample splitting. In other words, in these models efficient estimators can be constructed without sample splitting under the same conditions as those used for the construction with sample splitting. We demonstrate our results by constructing an efficient estimator using these ideas in a semiparametric additive regression model.

ABSTRACTIn this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n − 1 ∕ 2, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.

Semiparametric regression is a regression model consisting of two components, namely the parametric and the nonparametric component. The parametric component is a component with curves are known and the nonparametric component is a component with unknown shape of the curve. Bi-response semiparametric regression is regression analysis which has two response variables with the combination of parametric and nonparametric curves. The purpose of this research is to compare the bi-response semiparametric regression model using spline truncated with component parametric linear and quadratic. In this research, the method that used for the estimation parameter is WLS (Weighted Least Square) and the selection of optimal knot points is done by looking at the minimum GCV (Generalized Cross-Validation) value. WLS is minimizing partial derivative based on a parameter in the model. Each the linear bi-responsese semiparametric regression and quadratic bi-responsese semiparametric regression using spline truncated has two parameters that are parameter for parametric and nonparametric components. The difference between the two models contained in the parametric component of the predictor variable and parameter matrix of parametric component is estimator linear bi-response semiparametric regression model and is estimator quadratic bi-response semiparametric regression model using spline truncated.

The nonparametric binary logistic model assumes that the logit function is a linear function in the parameter, where the parameter depends on arbitrary fixed point. This study discusses the estimation of a nonparametric binary logistic regression model using the local likelihood logit estimation method. This method assumes that the log likelihood logit function depends on the multivariate kernel weighting. The parameter estimation of a nonparametric binary logistic regression model is obtained by maximizing the log likelihood logit function. The parameter estimation result are implicit, so to estimate it started with determining the optimal bandwidth value based on the maximum Cross Validation value. Furthermore, the optimal bandwidth value is used to estimate parameters using the multivariate Newton-Raphson algorithm until converging iterations are obtained. The parameter estimation process is done by creating a program in OSS-R software. This study also discusses the application of parametric and nonparametric binary logistic regression models in the case study of Type II Diabetes Mellitus patients at the Surabaya Hajj General Hospital. The modeling results show that the classification accuracy value of the parametric and nonparametric binary logistic regression models is 80.2% and 100% for the cut of value of 0.5.

Parameter estimates are proposed for semiparametric regression models with either linear or nonlinear parametric component, and arbitrarily many regressors in the nonparametric component. The estimates, which utilize nearest neighbour nonparametric regression, are shown to be consistent under mild conditions.

Recent research and studies show widespread interest in semiparametric regression model analysis, which combines parametric and nonparametric components. This interest is because it gives accurate and effective statistical model estimates. This paper proposes to improve estimates of semiparametric regression models using opposition-based learning technology on the golden Jackal Optimization algorithm to increase the accuracy of these models, accelerate convergence, and expand the exploration area. The effectiveness of using this algorithm was evaluated by comparing it with the original algorithm before optimization and the most commonly used methods for estimating the model statistically, such as CV and GCV. Using simulation, the results showed that the improvement in the OBL-GJO algorithm in terms of accuracy and convergence speed outperformed the original algorithm and traditional methods by a large margin in calculating the simulation results of the kernel semiparametric regression models. We strongly advocate for applying the GJO algorithm across various domains within machine learning, particularly in the realms of deep learning and reinforcement learning. Furthermore, we have employed enhanced and evolved algorithms to optimize semiparametric regression models effectively. To address the challenges encountered by any algorithm operating within a vast search landscape, we suggest an in-depth exploration of optimization techniques and integrating diverse algorithms, which could lead to more robust and efficient solutions.